Formation of a Differential Equation from a General Solution

Q. The differential equations satisfied by the system of parabolas $y^2=4a(x+a)$ is:

1. $y\left(\frac{dy}{dx}\right)+2x\left(\frac{dy}{dx}\right)-y=0$
2. $y{\left(\frac{dy}{dx}\right)}^2+2x\left(\frac{dy}{dx}\right)-y=0$
3. $y{\left(\frac{dy}{dx}\right)}^2-2x\left(\frac{dy}{dx}\right)-y=0$
4. $y{\left(\frac{dy}{dx}\right)}^2-2x\left(\frac{dy}{dx}\right)+y=0$

Q.

The differential equation for all the straight lines which are at a unit distance from the origin is

1. ${(y-x\frac{dy}{dx})}^2=1-{\left(\frac{dy}{dx}\right)}^2$

2. ${(y+x\frac{dy}{dx})}^2=1+{\left(\frac{dy}{dx}\right)}^2$

3. ${(y-x\frac{dy}{dx})}^2=1+{\left(\frac{dy}{dx}\right)}^2$

4. ${(y+x\frac{dy}{dx})}^2=1-{\left(\frac{dy}{dx}\right)}^2$

Q.

Differentiate $e^{ax}cos(bx+c).$

Q. The differential equation formed by eliminating parameters A and B from the family of curves given by$y=A{e}^{2x}+B{e}^{-2x}$ will be of ___ order.

1. First
2. Second
3. Third
4. None of the above

Q. The differential equation of all the straight lines which are at a constant distance of ${}^{\prime }{a}^{\prime }$ from the origin is

1. ${(y-x\left(\frac{dy}{dx}\right))}^2=a^2(1+{\left(\frac{dy}{dx}\right)}^2)$
2. ${(y+x\left(\frac{dy}{dx}\right))}^2=a^2(1+{\left(\frac{dy}{dx}\right)}^2)$
3. ${(y-x\left(\frac{dy}{dx}\right))}^2=a^2(1-{\left(\frac{dy}{dx}\right)}^2)$
4. ${(y+x\left(\frac{dy}{dx}\right))}^2=a^2(1-{\left(\frac{dy}{dx}\right)}^2)$

Q. If $ycosx+xcosy=\pi ,then{y}^{\prime \prime }\left(0\right)$ is

1. 1
2. $\pi$
3. \N
4. $-\pi$

Q. The differential equation representing the family of curves $y^2=2c(x+\sqrt{c})$ where c is a positive parameter, is of

1. Order 1
2. Order 2
   
3. Degree 3
4. Degree 4

Q. The differential equation whose solution is $(x-h{)}^2+(y-k{)}^2=a^2$ is (a is a constant)

1. ${[1+{\left(\frac{dy}{dx}\right)}^2]}^3=a^2\frac{d^{2}y}{d{x}^2}$
2. ${[1+{\left(\frac{dy}{dx}\right)}^2]}^3=a^2(\frac{d^{2}y}{d{x}^2}{)}^2$
3. ${[1+\left(\frac{dy}{dx}\right)]}^4=a^2(\frac{d^{2}y}{d{x}^2}{)}^2$
4. $Noneofthese$

Q. The differential equation whose general solution is given by,
$y=\left(c_{1}cos\right(x+c_2\left)\right)-\left(c_3{e}^{(-x+c_4)}\right)+\left(c_{5}sinx\right)$ where $c_1,c_2,c_3,c_4,c_5$ are arbitrary constants, is

1. $\frac{d^{4}y}{d{x}^4}-\frac{d^{2}y}{d{x}^2}+y=0$
2. $\frac{d^{3}y}{d{x}^3}+\frac{d^{2}y}{d{x}^2}+\frac{dy}{dx}+y=0$
3. $\frac{d^{5}y}{d{x}^5}+y=0$
4. $\frac{d^{3}y}{d{x}^3}+\frac{d^{2}y}{d{x}^2}+\frac{dy}{dx}-y=0$

Q.

Find the number of solutions of $5^x=x^2+x+1$.

___

Q. The degree of the differential equation satisfying the relation $\sqrt{1+x^2}+\sqrt{1+y^2}=\lambda (x\sqrt{1+y^2}-y\sqrt{1+x^2})$ is

1. 1
2. 2
3. 3
4. none of these

Q.

$\frac{d^{2}x}{d{y}^2}$ equals:

1. $-{\left(\frac{d^{2}y}{d{x}^2}\right)}^{-1}{\left(\frac{dy}{dx}\right)}^{-3}$

2. $\left(\frac{d^{2}y}{d{x}^2}\right){\left(\frac{dy}{dx}\right)}^{-2}$

3. $-\left(\frac{d^{2}y}{d{x}^2}\right){\left(\frac{dy}{dx}\right)}^{-3}$

4. ${\left(\frac{d^{2}y}{d{x}^2}\right)}^{-1}$

Q. The difference between degree and order of a differential equation that represents the family of curves given by $y^2=a(x+\frac{\sqrt{a}}{2}),a>0$ is

Q. The differential equation of the family of curves $y=A{e}^{3x}+B{e}^{5x}$, where A and B are arbitrary constants, is

1. $\frac{d^{2}y}{d{x}^2}+8\frac{dy}{dx}+15y=0$
   
2. $\frac{d^{2}y}{d{x}^2}-8\frac{dy}{dx}+15y=0$
   
3. $\frac{d^{2}y}{d{x}^2}-\frac{dy}{dx}+y=0$
4. $Noneofthese$

Q. S1: The differential equation of parabolas having their vertices at the origin and foci on the x-axis is an equation whose variables are separable
S2: The differential equation of the straight lines which are at a fixed distance p from the origin is an equation of degree 2
S3: The differential equation of all conics whose both axes coincide with the axes of coordinates is an equation of order 2

1. TTT
2. TFT
3. FFT
4. TTF

Q.

The differential equation corresponding to primitive $y=e^{dx}$ is

or

The elimination of the arbitrary constant m from the equation $y=e^{mx}$ gives the differential equation

[MP PET 1995, 2000; Pb. CET 2000]

1. $\frac{dy}{dx}=\left(\frac{y}{x}\right)logx$

2. $\frac{dy}{dx}=\left(\frac{x}{y}\right)logy$

3. $\frac{dy}{dx}=\left(\frac{y}{x}\right)logy$

4. $\frac{dy}{dx}=\left(\frac{x}{y}\right)logx$